Notes

One can consider so-called Anosov maps such as {x, y} -> Mod[m . {x, y}, 1] where m is a matrix such as {{2, 1}, {1, 1}}. Any initial condition containing only rational numbers will then yield repetitive behavior, much as in the shift map. But as soon as m itself contains rational numbers, complicated behavior can be obtained even with an initial condition such as {1, 1}.